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Universal Chern classes on the moduli of bundles

Donu Arapura

TL;DR

The paper addresses the problem of constructing universal cohomology classes on the moduli of bundles in the nonfine (noncoprime) setting, where a universal bundle may not exist. It introduces reduced Chern classes $\overline{c}_r(E)$, shows that certain invariants lift on the stable locus, and builds universal liftings $\tilde{c}_j$ in both complex analytic and étale cohomology using Brauer–Severi schemes and simplicial descent. The main result proves the existence of these universal reduced Chern classes on $C\times \mathfrak M_n^s(C)$ (and their stack-theoretic analogues), providing a robust framework to assign characteristic classes to moduli of bundles even when a universal bundle is absent. The work connects GIT moduli, Brauer–Severi geometry, simplicial methods, and stack cohomology to yield a coherent interpretation of universal cohomology on moduli of bundles with fixed determinant.

Abstract

The goal of this paper is to construct universal cohomology classes on the moduli space of stable bundles over a curve when it is not a fine moduli space, i.e. when the rank and degree are not coprime. More precisely, we show that certain Chern classes of the universal bundle on the product of the curve with the moduli stack of bundles lift to the product of the curve with the moduli space of stable bundles.

Universal Chern classes on the moduli of bundles

TL;DR

The paper addresses the problem of constructing universal cohomology classes on the moduli of bundles in the nonfine (noncoprime) setting, where a universal bundle may not exist. It introduces reduced Chern classes , shows that certain invariants lift on the stable locus, and builds universal liftings in both complex analytic and étale cohomology using Brauer–Severi schemes and simplicial descent. The main result proves the existence of these universal reduced Chern classes on (and their stack-theoretic analogues), providing a robust framework to assign characteristic classes to moduli of bundles even when a universal bundle is absent. The work connects GIT moduli, Brauer–Severi geometry, simplicial methods, and stack cohomology to yield a coherent interpretation of universal cohomology on moduli of bundles with fixed determinant.

Abstract

The goal of this paper is to construct universal cohomology classes on the moduli space of stable bundles over a curve when it is not a fine moduli space, i.e. when the rank and degree are not coprime. More precisely, we show that certain Chern classes of the universal bundle on the product of the curve with the moduli stack of bundles lift to the product of the curve with the moduli space of stable bundles.
Paper Structure (7 sections, 11 theorems, 69 equations)

This paper contains 7 sections, 11 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Some background on moduli spaces
  3. Reduced Chern classes
  4. Reduced Chern classes for Brauer-Severi schemes
  5. Chern classes of bundles on simplicial schemes
  6. Main theorem
  7. Interpretation using stacks

Key Result

Proposition 2.1

There exists a Brauer--Severi scheme $\mathfrak P$ over $C\times \mathfrak M_n^s(C,L)$, such that $\mathbb{P}(\mathcal{E})=p^{*}\mathfrak P$.

Theorems & Definitions (19)

  • Proposition 2.1: bbgn
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 9 more